Any physical channel of communication offers two potential reasons why its
capacity (the number of bits it can transmit in a unit of time) might be
unbounded: (1) Infinitely many choices of signal strength at any given instant
of time, and (2) Infinitely many instances of time at which signals may be
sent. However channel noise cancels out the potential unboundedness of the
first aspect, leaving typical channels with only a finite capacity per instant
of time. The latter source of infinity seems less studied. A potential source
of unreliability that might restrict the capacity also from the second aspect
is delay: Signals transmitted by the sender at a given point of time may not be
received with a predictable delay at the receiving end. Here we examine this
source of uncertainty by considering a simple discrete model of delay errors.
In our model the communicating parties get to subdivide time as microscopically
finely as they wish, but still have to cope with communication delays that are
macroscopic and variable. The continuous process becomes the limit of our
process as the time subdivision becomes infinitesimal. We taxonomize this class
of communication channels based on whether the delays and noise are stochastic
or adversarial; and based on how much information each aspect has about the
other when introducing its errors. We analyze the limits of such channels and
reach somewhat surprising conclusions: The capacity of a physical channel is
finitely bounded only if at least one of the two sources of error (signal noise
or delay noise) is adversarial. In particular the capacity is finitely bounded
only if the delay is adversarial, or the noise is adversarial and acts with
knowledge of the stochastic delay. If both error sources are stochastic, or if
the noise is adversarial and independent of the stochastic delay, then the
capacity of the associated physical channel is infinite
